# Signals Cross Correlation from Their Power Spectral Densities?

For two real valued signals $$f$$ and $$g$$ with their corresponding power spectral densities $$|F(k)|^2$$ and $$|G(k)|^2$$ Is there a way to get an intuition of how correlated $$f$$ and $$g$$ are just by looking at the power spectral densities (PSDs)? For example does the fact that the PSDs of these two signals are so similar imply that they are cross correlated?

So far I have considered that $$|F(k)|^2 = F(k) \overline{F(k)}$$ $$|G(k)|^2 = G(k) \overline{G(k)}$$

and the cross correlation we want to find is

$$F[f \star g] = F(k)\overline{G(k)} = G(-k)\overline{F(-k)}$$

and so it seems that there could be a way to get an exact result for the cross correlation from the PSDs, however I would also be happy just to get a hand wavy idea of if these two signals are cross correlated

No,

For instance 2 instances of AWGN with the same STD have the same PSD while being totally uncorrelated (Moreover, Statistically Independent).

Also, pay attention that Correlation is not commutative:

$$f \ast g \neq g \ast f \Leftrightarrow F \left( k \right) \overline{G \left( k \right)} \neq G \left( k \right) \overline{F \left( k \right)}$$

• Yes thank you, that's a mistake, I think this edited version holds for real valued signals – Stephen Jackson Jun 3 '19 at 8:35
• Leave the Math a side, I gave you a counter example why you can't infer much. – David Jun 3 '19 at 8:41
• I understand this but is this not a special case? If your signals are not randomly distributed noise (which mine aren't) surely the spectral composition must be an indicator. If you have a very dominant frequency sinusoid in two signals even if theyre out of phase that must imply some cross correlation since the two sinusoids themselves would definitely be cross correlated – Stephen Jackson Jun 3 '19 at 8:44
• If you have more information about your specific signals you may derive something. But in general no. Actually the Harmonic Signals are great example. Sine and Cosine will look the same but they are, again, un correlated. Phase is really important and probably you can decorrelate any signal using it. – David Jun 3 '19 at 8:52
• Not correlated, but I think that sine and cosine might be cross-correlated – Stephen Jackson Jun 3 '19 at 9:02